In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by:

+

For local equilibrium, we are able to find the mean velocity profile <math>u</math> from the turbulent kinetic energy <math>k</math> (equation 4) and the mixing length <math>l_m</math> (equation 5), by:

Two-equation - eddy viscosity model

One-equation eddy viscosity model

Algebraic eddy viscosity model

(3)

is the mixing length.

Algebraic model for the turbulent kinetic energy

(4)

is the shear velocity and a model parameter.

For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled -equation [Nezu and Nakagawa (1993)] obtained a similar semi-theoretical equation.

Algebraic model for the mixing length

For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [Absi (2006)]:

(5)

, is the hydrodynamic roughness.
For a smooth wall ():

(6)

the algebraic eddy viscosity model is therefore

(7)

The mean velocity profile

For local equilibrium, we are able to find the mean velocity profile from the turbulent kinetic energy (equation 4) and the mixing length (equation 5), by:

(8)

Figure (1) shows that the velocity profile obtained from equations (8), (4) and (5) (solid line) is more accurate than the logarithmic velocity profile (dash-dotted line).